Problem 74 - College Physics

Question

Disks \(A\) and \(B\) are mounted on shaft \(S S\) and may be connected or dis- connected by clutch \(C .\) (See Figure \(10.82 . )\) Disk \(A\) is made of a lighter material than disk \(B,\) so the moment of inertia of disk \(A\) about the shaft is one-third that of disk \(B\) . The moments of inertia of the shaft and clutch are negligible. With the clutch disconnected, \(A\) is brought up to an angular speed \(\omega_{0 . \text { . The }}\) accelerating torque is then removed from \(A,\) and \(A\) , and \(A\) is coupled to disk \(B\) by the clutch. (You can ignore bearing friction.) It is found that 2400 \(\mathrm{J}\) of thermal energy is developed in the clutch when the connection is made. What was the original kinetic energy of disk \(A\) ?

Step-by-Step Solution

Answer

The initial kinetic energy of disk A was 2400 J.

Step 1: Identify Parameters and Relations

Let the moment of inertia for disk B be \( I_B \). Since A's moment is one-third of B's, \( I_A = \frac{1}{3} I_B \). With initial angular speed \( \omega_0 \) for A, its initial kinetic energy is \( KE_A = \frac{1}{2} I_A \omega_0^2 \).

Step 2: Calculate Total Moment of Inertia After Coupling

After coupling, both disks combine. The combined moment of inertia is \( I_{total} = I_A + I_B = \frac{1}{3} I_B + I_B = \frac{4}{3} I_B \).

Step 3: Apply Conservation of Angular Momentum

Before coupling, only A has momentum: \( L_{initial} = I_A \omega_0 \). After coupling, the system's momentum must be: \( L_{final} = I_{total} \omega_f \). Set these equal: \( I_A \omega_0 = I_{total} \omega_f \). Solve for \( \omega_f \): \( \omega_f = \frac{I_A \omega_0}{I_{total}} = \frac{\left(\frac{1}{3} I_B\right) \omega_0}{\frac{4}{3} I_B} = \frac{1}{4} \omega_0 \).

Step 4: Determine Final Kinetic Energy

The final kinetic energy of the system after coupling is \( KE_f = \frac{1}{2} I_{total} \omega_f^2 = \frac{1}{2} \cdot \frac{4}{3} I_B \cdot \left(\frac{1}{4} \omega_0\right)^2 = \frac{1}{2} \cdot \frac{4}{3} I_B \cdot \frac{1}{16} \omega_0^2 = \frac{1}{24} I_B \omega_0^2 \).

Step 5: Calculate the Initial Kinetic Energy of Disk A

Thermal energy developed, \( E_{thermal} = 2400 \) J, is the difference between initial and final kinetic energies. Therefore, \( KE_A - KE_f = 2400 \). Plug in expressions, \( \frac{1}{2} I_A \omega_0^2 - \frac{1}{24} I_B \omega_0^2 = 2400 \). Since \( I_A = \frac{1}{3} I_B \), the equation becomes \( \frac{1}{2} \cdot \frac{1}{3} I_B \omega_0^2 - \frac{1}{24} I_B \omega_0^2 = 2400 \). Simplify to find \( KE_A = \frac{1}{6} I_B \omega_0^2 = 2400 \). Thus, \( KE_A = 2400 \) J.

Key Concepts

Moment of Inertia

Moment of inertia is a crucial concept in rotational dynamics. It measures an object's resistance to changes in its rotation. Think of it as the rotational equivalent of mass in linear motion.

Just like how heavier objects are harder to accelerate, objects with larger moments of inertia are harder to spin. For example, in the problem above, disk A has less moment of inertia compared to disk B, as it is made from a lighter material. Specifically, disk A has one-third the moment of inertia of disk B. This makes disk A easier to bring up to speed than disk B.

• The formula for the moment of inertia depends on the object's shape and the axis about which it rotates.
• For simple shapes like disks, there are standard formulas.
• In this case, the problem focuses mainly on comparing moment of inertia between the two disks.

Understanding moment of inertia helps us predict how rotation will behave when forces act upon it.

Conservation of Angular Momentum

Conservation of angular momentum is a key principle in rotational dynamics. It states that if no external torque acts on a system, its angular momentum remains constant. This is analogous to how, in linear motion, momentum is conserved when no external forces are involved.

In the problem, when disk A couples with disk B, the system's angular momentum remains conserved. Initially, disk A is the only one rotating, with a momentum given by its moment of inertia and angular speed. Once it connects with disk B, they rotate as a single system.

• Initial angular momentum: Calculated as the product of disk A's moment of inertia and angular speed.
• Final angular momentum: Calculated with the combined moment of inertia and the new angular speed of the system.
• Conservation equation: Ensures both calculations equal each other, adjusting speeds accordingly.

This principle allows us to equate the initial and final angular momentum to find unknown values, such as the final angular speed after the two disks couple.

Kinetic Energy

Kinetic energy in rotational motion depends on both the moment of inertia and the square of the angular velocity. It's the energy due to motion, just like how throwing a ball gives it kinetic energy because of its movement.

In this exercise, the initial kinetic energy was found by calculating how fast disk A was spinning before connecting with disk B. After coupling, the new system's kinetic energy was computed using the new angular speed, which had decreased due to the conservation of angular momentum.

• Initial kinetic energy: Calculated using disk A's moment of inertia and initial angular speed.
• Final kinetic energy: Found by recalculating after disk A and B are combined.
• Energy change: The difference between these energies, explaining how much energy became thermal.

Using these calculations, the initial kinetic energy of disk A can be determined by considering the energy changes during the coupling.

Thermal Energy

Thermal energy is often a byproduct in energy transformations when friction or other forces are involved. In this exercise, thermal energy is generated in the clutch connecting disks A and B during the coupling process.

When disk A couples with disk B, some of the initial kinetic energy is converted into thermal energy due to the friction within the clutch. This conversion is why the system’s total initial kinetic energy reduces.

• Thermal energy generated: Quantified as 2400 J in this problem.
• Cause of thermal energy: Primarily due to friction and the conversion of some rotational kinetic energy.
• Energy conservation: The total energy must balance, leading to the difference between initial and final kinetic energies being equal to the thermal energy generated.

Understanding how thermal energy is produced helps us grasp the broader picture of energy conservation and transformation during mechanical processes.